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The Astrophysical Journal, 763:10 (13pp), 2013 January 20 doi:10.1088/0004-637X/763/1/10C© 2013. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

ENHANCED MAGNETIC COMPRESSIBILITY AND ISOTROPIC SCALE INVARIANCEAT SUB-ION LARMOR SCALES IN SOLAR WIND TURBULENCE

K. H. Kiyani1,2, S. C. Chapman2, F. Sahraoui3, B. Hnat2, O. Fauvarque1,4, and Yu. V. Khotyaintsev51 Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK; [email protected]

2 Centre for Fusion, Space and Astrophysics, University of Warwick, Coventry CV4 7AL, UK3 Laboratoire de Physique des Plasmas, Observatoire de Saint-Maur, F-94107 Saint-Maur-Des-Fosses, France

4 Departement de Physique, Ecole Normale Superieure, 24 rue Lhomond, F-75005 Paris, France5 Swedish Institute of Space Physics, SE-75121 Uppsala, Sweden

Received 2012 July 21; accepted 2012 November 11; published 2012 December 27

ABSTRACT

The anisotropic nature of solar wind magnetic turbulence fluctuations is investigated scale by scale using highcadence in situ magnetic field measurements from the Cluster and ACE spacecraft missions. The data span fivedecades in scales from the inertial range to the electron Larmor radius. In contrast to the inertial range, there is asuccessive increase toward isotropy between parallel and transverse power at scales below the ion Larmor radius,with isotropy being achieved at the electron Larmor radius. In the context of wave-mediated theories of turbulence,we show that this enhancement in magnetic fluctuations parallel to the local mean background field is qualitativelyconsistent with the magnetic compressibility signature of kinetic Alfven wave solutions of the linearized Vlasovequation. More generally, we discuss how these results may arise naturally due to the prominent role of the Hall termat sub-ion Larmor scales. Furthermore, computing higher-order statistics, we show that the full statistical signatureof the fluctuations at scales below the ion Larmor radius is that of a single isotropic globally scale-invariant processdistinct from the anisotropic statistics of the inertial range.

Key words: magnetic fields – methods: data analysis – plasmas – solar wind – turbulence

Online-only material: color figures

1. INTRODUCTION

In situ measurements of fields and particles in the interplane-tary solar wind provide unique observations for the study of a tur-bulent plasma (Tu & Marsch 1995; Horbury et al. 2005; Bruno& Carbone 2005). As in neutral fluid turbulence, an inertialrange of magnetohydrodynamic (MHD) turbulence is indicatedby the observations of a power spectral density (PSD) exhibitinga power-law form over a large range of scales. This power law isa manifestation of a scale-invariant turbulent cascade of energyfrom large to small scales. Unlike neutral fluids where dissipa-tion is carried out by viscosity, interplanetary space plasmas arevirtually collisionless. On spatial scale characteristic of the ionsthere is a transition in the PSD from the inertial range at lowerfrequencies to a steeper power law spanning up to two decadesin scale to the electron scales (Sahraoui et al. 2009; Kiyani et al.2009b). This second interval of scaling in the PSD was dubbedthe dissipation range in analogy with hydrodynamic turbulence(Leamon et al. 1999). The nature of the fluctuations on thesekinetic scales, the mechanisms by which the turbulent energyis cascaded and dissipated, and the possible role of dispersivelinear wave modes (Leamon et al. 1999; Bale et al. 2005; Garyet al. 2008, 2010) are all hotly debated as vital ingredients ofany future model of this dissipation range (Schekochihin et al.2009; Alexandrova et al. 2008; Howes et al. 2008; Sahraouiet al. 2010).

Anisotropy, with respect to the background magnetic field, is acentral feature of plasma turbulence in the solar wind (Matthaeuset al. 1996; Osman & Horbury 2007; Narita et al. 2010; Sahraouiet al. 2010), with the fluctuating components transverse andparallel to the background magnetic field displaying manifestdifferences in both their dynamics and statistics. The seminalstudy of Belcher & Davis (1971) used Mariner 5 observations to

investigate the field variance tensor projected onto an orthonor-mal “field-velocity” coordinate system. They found that themajority of the power is in the transverse fluctuations with aratio between transverse and parallel (compressible) compo-nents ∼9:1. As first seen in high-cadence WIND spacecraft data(Leamon et al. 1998b), this ratio decreases to ∼5:1 in the dissi-pation range, indicating enhanced magnetic compressible fluc-tuations in this sub-ion Larmor scale range. This result showingan enhanced level of magnetic compressibility was repeatedwith a large set of ACE spacecraft data intervals (Hamilton et al.2008) and also in much higher cadence data from the Clusterspacecraft (Alexandrova et al. 2008).

In this article, we conduct a scale by scale study of theanisotropy in transverse and parallel fluctuations over fivedecades in temporal scales from a few hours to tens of mil-liseconds. Our background magnetic field is scale dependent,computed self-consistently with the scale-dependent fluctua-tions as in Horbury et al. (2008) and Podesta. (2009), i.e., alocal—as opposed to global—background magnetic field. Weshow that the findings of Leamon et al. (1998b) and Alexandrovaet al. (2008) are in fact the result of a successive scale-invariantreduction in the power ratio between the two components aswe move to smaller scales in the dissipation range. This re-duction results in isotropy between all three components ofmagnetic field fluctuations (two transverse and one parallel) atthe electron Larmor radius ρe, i.e., equipartition of magneticenergy between all three magnetic field components. Using aHall-MHD model we provide a simple description of how theHall term in the generalized Ohm’s law is responsible for therise of parallel (compressible) magnetic fluctuations and the re-sultant isotropy at scales around ρe. In addition, by comparingwith linear kinetic solutions of the Vlasov equation and com-puting the magnetic compressibility (Gary & Smith 2009) we

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The Astrophysical Journal, 763:10 (13pp), 2013 January 20 Kiyani et al.

show that this scale-invariant reduction in the power ratio isalso qualitatively consistent with the transition from shear orhighly oblique Alfvenic fluctuations in the inertial range pos-sessing no magnetic compressibility to kinetic Alfven-wave-likefluctuations which rapidly develop magnetic compressible com-ponents at scales around the ion Larmor radius (Gary & Smith2009; Sahraoui et al. 2012). This nature of the magnetic com-pressibility is also suggested in the recent work by Salem et al.(2012) who, using a global background field, also computedthe magnetic compressibility and compared it with wave solu-tions of the linearized warm two-fluid equations. Importantly,by calculating higher order statistics and the probability densityfunctions (PDFs) we show for the first time that the dissipationrange magnetic fluctuations are characterized statistically by asingle isotropic signature; in stark contrast to the anisotropicinertial range.

We stress that the anisotropy discussed in this article is inthe full vector direction of field fluctuations, δB, projectedparallel and transverse to the local magnetic field direction(also known as variance anisotropy; Belcher & Davis 1971).This is in contrast to the anisotropy measured in the vectorr which characterizes the spatial scale of the fluctuations (orwavevector k in Fourier space). In the context of the latteranisotropy, the spatial variations in the magnetic field that wemeasure are a sum of all the components of k which are projectedalong the spacecraft trajectory through the solar wind, i.e.,along the solar wind velocity unit vector −Vsw; these are thenDoppler shifted (via the Taylor frozen-in-flow hypothesis) tospacecraft frame frequencies, f, resulting in a reduced spectrumP (f, θBV ) = ∫

d3kP (k)δ(2πf − k · Vsw), where the δ functionis doing the “reducing,” and θBV shows the explicit dependenceof the reduced spectra on the angle between the backgroundmagnetic field and the bulk solar wind velocity vectors (seeForman et al. 2011 and Fredricks & Coroniti 1976 for furtherdetails). The full k spectrum, P (k), can only be measured bymultispacecraft techniques which take into account the fullthree-dimensional spatial variation of the magnetic field, e.g.,k-filtering (Pincon & Lefeuvre 1991). In the context of thereduced spectrum, the magnetic field observations investigatedin this article are dominated by wavevector components whichare strongly oblique (70◦–90◦) to the background magnetic field,as indicated by the angle between the background magneticfield and solar wind velocity vectors. In such intervals, theinertial range is ubiquitously seen to have a spectral index of∼−5/3, while in the dissipation range this steepens to ∼−2.8(Kiyani et al. 2009b; Alexandrova et al. 2009; Chen et al. 2010;Sahraoui et al. 2010). Also, for such intervals, multispacecraftobservations of the full P (k) (Narita et al. 2010; Sahraouiet al. 2010) suggest that most of the power is in the k⊥ ratherthan the k‖ components, suggesting that highly oblique angledcomponents of k are very much representative of most of themagnetic field fluctuation power.

A brief synopsis of the paper is as follows: Section 2 de-scribes the spacecraft data and summarizes the plasma param-eters involved, as well as the analysis methods used; Section 3forms the bulk of the paper and is split between (1) a study anddiscussion of the rising magnetic compressibility and compo-nent anisotropy in the dissipation range and (2) an anisotropicstudy of the higher-order statistics. Finally, Section 4 summa-rizes our findings and brings together the insights obtained fromthese results, concluding with an outlook for future investiga-tions. Details of the undecimated discrete wavelet transform(UDWT) that we use to decompose the magnetic field into

scale-dependent background field and fluctuations, for use inthis study, are described in the Appendix.

2. OBSERVATIONS AND METHODS

We discuss two near-similar intervals of quiet ambient solarwind from observations of the Cluster and ACE spacecraftmissions (Escoubet et al. 1997; Garrard et al. 1998; all inGSE coordinates). The Cluster (spacecraft 4) interval at 450 Hzcadence (same interval as in Kiyani et al. 2009b) is of an hourduration 2007/01/30 00:10–01:10 UT when the instrumentswere operating at burst mode and will primarily be used tostudy the dissipation range at spacecraft frequencies above 1 Hz.We construct a combined data set from the DC magnetic field(sampled at 67 Hz) of the flux gate magnetometer (FGM) forfrequencies below 1 Hz, with the high-frequency (oversampledat 450 Hz) search-coil magnetometer data from the STAFF-SCM experiment for frequencies above 1 Hz, using the waveletreconstruction method (Alexandrova et al. 2004; Chen et al.2010). The ACE magnetometer (MAG) interval at 1 Hz cadenceis over two days 2007/01/30 00:00 UT to 2007/01/31 23:59 UTand samples the inertial range at spacecraft frequencies below1 Hz. The ACE interval was only needed in order to obtain betterestimates of the higher-order statistics in the inertial range (asthe Cluster FGM interval is relatively short for this), and thuswill only be restricted to this study in the penultimate section ofthe paper; for the other studies of the inertial range, the FGMdata interval was used.

Both these intervals are in stationary fast wind (�667 km s−1)with similar plasma parameters and are sufficiently large datasets so that the sample size variance errors in the computedstatistics are negligible (Kiyani et al. 2009a). The followingplasma quantities are from the Cluster FGM, CIS, PEACE, andWHISPER instruments: average magnetic field B � 4.5 nT,electron plasma βe � 1.2, plasma density ne � 4 cm−3,Alfven speed VA � 50 km s−1, perpendicular ion temperatureTi⊥ � 24 eV, electron temperature Te � 22 eV, and ionand electron Larmor radii ρi � 110 km and ρe � 1.7 km,respectively. The interval is free from any ion or electronforeshock effects at Earth’s bow shock. Moreover, using high-resolution proton moments computed from the 3DP instrument(Lin et al. 1995) on board the WIND spacecraft at 1 AU andat the same interval as Cluster, the proton plasma parameters(parallel and perpendicular proton temperatures Tp‖ � 26 eVand Tp⊥ � 34 eV, parallel proton plasma βp‖ � 1.2) indicatethat the intervals are stable to proton pressure anisotropy-driven instabilities (Bale et al. 2009). From the ACE SWEPAMinstrument the ion plasma beta βi � 1.5 shows that the ion-inertial length λi � ρi .

There is a growing opinion (Chapman & Hnat 2007; Horburyet al. 2008; Podesta. 2009; Luo & Wu 2010) that it is a localscale-dependent mean field consistent with the scale-dependentfluctuations, rather than a large-scale global field, which shouldbe used in studies of anisotropic plasma turbulence. This isnot only self-consistent but also ensures that there are no sig-nificantly large spectral gaps between the frequencies of thefluctuations being studied and the mean fields being projectedupon. In accordance with this approach, we use the UDWTmethod to decompose the fields into local scale-dependent back-ground magnetic fields and fluctuations, B(t, f ) and δB(t, f ),respectively, where f explicitly shows the frequency or scaledependence—the parallel and transverse fluctuations,δB(‖/⊥)(t, f ), are then obtained from these. The background

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−8

−6

−4

−2

0

2

log 10

PS

D [n

T2 H

z−1 ]

−2 −1 0 1

−1.2

−1

−0.8

−0.6

−0.4

log10

spacecraft frame frequency [Hz]

log 10

PS

D||/P

SD

⊥

paralleltransversesearch coil noise

isotropy

Figure 1. Upper panel: PSD (from Cluster) of the transverse and parallelcomponents spanning the inertial and dissipation ranges. Standardized samplesize variance errors are smaller than the markers. The search-coil sensitivityfloor PSD is obtained from the z-component (spacecraft SR2 coordinates) ofa very quiet period in the magnetotail lobes (2007/06/30 15:00–15:05 UT)as a proxy for the instrumentation noise. Lower panel: ratio of parallel overtransverse PSDs. Horizontal dot-dashed line indicates a ratio of 1/2 whereisotropy in power occurs. Vertical dashed and dash-dotted lines indicate ρi

and ρe , respectively, Doppler-shifted to spacecraft frequency using the Taylorhypothesis. The little blip in PSD‖ at ∼0.25 Hz (and in PSD‖/PSD⊥) is due tothe residual spacecraft (∼4 s) spin tone in the FGM signal; it is more noticeablein PSD‖ due to the lower power in parallel fluctuations in the inertial range.

(A color version of this figure is available in the online journal.)

to this particular wavelet method and brief details of the algo-rithms are described in the Appendix.

3. RESULTS AND DISCUSSIONS

3.1. Power Isotropy and Enhanced Magnetic Compressibility

In keeping with Parseval’s theorem for the conservation of theL2-norm (energy conservation) the wavelet PSD for the paralleland transverse magnetic field components is given by

PSD‖(⊥)(f ) = 2ΔN

N∑j=1

δB2‖(⊥)(tj , f ) , (1)

where δB⊥(tj , f ) =√

δB2⊥1(tj , f ) + δB2

⊥2(tj , f ) is the magni-tude of the total transverse fluctuations at time tj and frequencyf, Δ is the sampling period between each measurement, and Nis the sample size at each frequency f. For the Cluster intervalthe PSD‖ and PSD⊥ are shown in Figure 1. The spectral indicesobtained are �−1.62±0.01 and �−1.59±0.01 for parallel andtransverse components, respectively, in the inertial range; and�−2.67 ± 0.01 and �−2.94 ± 0.01 for parallel and transverse

components respectively, in the dissipation range. The lowerpanel of Figure 1 shows that not only do these results recoverthe ∼9:1 anisotropy ratio of Belcher & Davis (1971), they alsoshow that the decrease in the anisotropy observed by Leamonet al. (1998b), Alexandrova et al. (2008), and Hamilton et al.(2008) in the dissipation range is actually a scale-free progres-sion to isotropy. This progression of the anisotropy in the powerratio PSD‖/⊥ (a measure of magnetic compressible fluctuations)begins at the spectral break (spacecraft frequency ∼0.25 Hz),just before the calculated ρi , and follows the power-law rela-tionship PSD‖/⊥ ∼ f 1/3±0.05 to ρe, where isotropy in powerbetween the three components (one parallel and two transversecomponents) is achieved. This isotropy corresponds to a valueof PSD‖/PSD⊥ = 1/2 and is indicated in the lower panel ofFigure 1. Although this enhancement of parallel, or compress-ible, fluctuations in the dissipation range has already been com-mented upon by various authors (Leamon et al. 1998b; Hamiltonet al. 2008; Alexandrova et al. 2008), it is normally shown tobe nearly constant (apart from in Salem et al. 2012). To ourknowledge, this is the first time that an observation of isotropyhas been noted to occur at kρe � 1, although it is also stronglysuggested in the PSDs in Sahraoui et al. (2010).

The lower panel of Figure 1 is calculated from the ratio ofthe averages of the parallel and transverse fluctuations to showa measure of the anisotropy. As such it does not constitute aproper estimate of the ensemble average of the anisotropy. Aproper measure would be to take the ratio of δB2

‖ (tj , f ) andδB2

⊥(tj , f ) at each time tj, and then average over this ensembleof realizations of the anisotropy. However, this is prone tolarge errors induced by very large spikes caused by purely, ornearly pure, parallel fluctuations which result in divisions by avery small number. To overcome this problem and to obtain aproper ensemble averaged anisotropy measure, we compute analternative and robust metric of the magnetic compressibility(similar in form to the expressions in Gary & Smith 2009 andAlexandrova et al. 2008) defined as

C‖(f ) = 1

N

N∑j=1

δB2‖ (tj , f )

δB2‖ (tj , f ) + δB2

⊥(tj , f ), (2)

i.e., the compressibility is calculated locally from the time-dependent fluctuations and then averaged. Converting the space-craft frequency into wavenumber using the Taylor hypothesisand normalizing by the averaged ion gyroradius for the in-terval, Figure 2 shows C‖(kρi) computed using the localscale-dependent mean field. We have binned the latter into 10◦angle bins (angle between Vsw and e‖(tj , f )) to show whichcomponents of the k variation we are measuring with respectto the local background (scale-dependent) magnetic field. Onthe same plot, and for comparison, we have also included thecalculation of the magnetic compressibility using a global back-ground magnetic field—the latter consists of the mean averageof the magnetic field vector over the whole interval being stud-ied. The angle of the solar wind velocity to the backgroundmagnetic field using this global mean field would correspondto wavevector components at an angle of ∼75◦ to the back-ground field, both within the inertial and dissipation ranges.From these plots of C‖(kρi) we can see that there is not only alarge difference between C‖ calculated using a local as opposedto a global field, there is also no significant difference betweenC‖ calculated using a local field for the separate k-componentangle bins. Also, similar to the lower panel of Figure 1, inall the curves we again see the enhancement of the magnetic

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Mag

netic

Com

pres

sibi

lity

C||

local field 55º-65ºlocal field 65º-75ºlocal field 75º-85ºlocal field 85º-95ºglobal fieldlinear Vlasov 85ºlinear Vlasov 89.99º

isotropy

Figure 2. Magnetic compressibility C|| (from Cluster), computed as inEquation (2), against the normalized wavenumber for both local (scale-dependent) and global (computed over the whole interval) background magneticfields. The normalized wavenumber was computed assuming the Taylor hypoth-esis. Numerical solutions of the linearized Vlasov equation, corresponding tothe kinetic Alfven wave mode (computed using the solar wind parameters forthis interval), are also shown in comparison to illustrate the rising magneticcompressibility possessed by these modes at the ion gyroscale. The horizon-tal dashed line at C|| = 1/3 indicates the level at which full isotropy of themagnetic field fluctuations occurs. As in Figure 1, the statistical sample sizevariance errors are smaller than the markers, and the effect of the spacecraftspin is clearly seen at kρi ∼ 0.25.


compressibility—with isotropy between all three components ofthe vector magnetic field fluctuations being reached at kρe � 1.

Lastly, one should note from Figure 2 that the differencebetween the magnetic compressibility computed with global andlocal scale-dependent background magnetic fields decreases aswe move to smaller scales. At a first glance this is a non-intuitiveresult, as we would expect that the local field increasinglymimics the global field at larger scales. However, this can beeasily resolved by the fact that if the fluctuations are isotropicallydistributed, it does not matter which basis one is projecting thefluctuations on—they will look the same in all bases. So if thefluctuations are becoming more isotropic toward the smallerscales, the difference between the global and local backgroundmagnetic field bases will be less.

3.1.1. Rising Magnetic Compressibility in Kinetic Alfven WaveSolutions of the Linearized Vlasov Equation

Theories of plasma turbulence which advocate that a turbu-lent energy cascade within the dissipation range is mediated bythe various linear wave modes of a plasma (Leamon et al. 1999;Schekochihin et al. 2009; Gary et al. 2008, 2012; Chang et al.2011), suggest predictions for C‖ (Gary & Smith 2009; Sahraouiet al. 2012; TenBarge et al. 2012). As the wave modes advocatedto cascade energy at these scales are dispersive in nature (kineticAlfven and Whistler waves), many of the proponents of suchtheories suggest that what has been dubbed the dissipation rangefor purely historical reasons, should actually be called the dis-persive range (Stawicki et al. 2001; Sahraoui et al. 2012). Withinthe context of sub-ion Larmor scales, there is a growing body ofwork showing that the fluctuations at these scales share the char-acteristic of kinetic Alfven waves (Leamon et al. 1998b, 1999;Bale et al. 2005; Howes et al. 2008; Sahraoui et al. 2009, 2010;Salem et al. 2012). Indeed, the enhancement of the magneticcompressibility seen above is strongly consistent with the tran-

sition, around kρi ∼ 1, of purely incompressible shear Alfvenwaves into kinetic Alfven waves, with a strong compressivecomponent (Hollweg 1999) and propagating at highly oblique(near-perpendicular) angles to the background magnetic field.

In Figure 2, we overlay the observational results of C‖by predictions of C‖(kρi) from numerical solutions of thelinearized Vlasov equation for the kinetic Alfven wave modeat a highly oblique (85◦) angle and at a virtually perpendicular(89.◦99) angle, using the same bulk plasma field and particleparameters of our data. These numerical solutions were plottedusing the WHAMP code (Ronnmark 1982) which assumes astrong guide field, i.e., a global background magnetic field. Theplot shows that our results are qualitatively consistent with C‖predicted for kinetic Alfven waves. This enhancement of thecompressive component in the kinetic Alfven wave is due tothe coupling of the Alfven mode to the purely compressiveion acoustic mode (Leamon et al. 1999). In the inertial rangebelow kρi ∼ 1 both of these modes are decoupled (Howeset al. 2006; Schekochihin et al. 2009). In the context of suchtheories, the 10% compressible component that we see in theinertial range is most likely due to the presence of magnetosoniccompressible modes in the data, all at highly oblique (nearlyperpendicular) angles to the background magnetic field. Themodes can be in the form of slow modes, fast modes or non-propagating (ω = 0) pressure-balanced structures advectedby the flow (mirror modes), all of which have purely parallel(compressible) fluctuations (Sahraoui et al. 2012). Although arecent paper by Howes et al. (2011a) shows that slow modesor pressure-balanced structures constitute the dominant partof the compressible fluctuations in the inertial range, fastmagnetosonic modes may also exist. This can be supportedby the fact that these fast modes are shown to be considerablyundamped (compared to slow modes) in the linear theory athighly oblique angles of propagation and high plasma beta(βi � 1; Sahraoui et al. 2012), conditions very relevant tothe solar wind. In Figure 2, the departure of our results fromthe theoretical solution at kρi < 1 can be explained by the factthat the theoretical curve is only for kinetic Alfven waves—anaddition of purely compressible magnetosonic modes might beable to reduce the difference significantly (as in TenBarge et al.2012). In addition, the theoretical plot is calculated assuminga strong global background magnetic field. Importantly, thetheoretical plot for kinetic Alfven waves shows that the magneticcompressibility should start to rise before kρi ∼ 1 which is alsowhat our results show. Recently, Salem et al. (2012) showed asimilar result for magnetic compressibility (their definition ofC‖ not being a squared quantity as in this article) with betteragreement with the theoretical C‖ curve for kinetic Alfven wavesthan the results presented here. The differences in our resultsand those of Salem et al. (2012) could be due to the intervalsanalyzed: our interval is fast wind with βi ∼ 1.5, whereas theirinterval is slow wind with βi ∼ 0.5; that they used two-fluidlinear warm plasma relations, instead of linear Vlasov solutions;or their use of a global background magnetic field instead of alocal one. To put both these results in their proper place requires alarge data survey with intervals that sample more comprehensiveplasma conditions, as well as taking into account details such asthe linear polarization property of kinetic Alfven waves.

3.1.2. The Role of the Hall Term in the Enhancement of MagneticCompressibility and Component Isotropy

In the more general nonlinear setting, this enhancement ofmagnetic compressible fluctuations can be explained by the

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increased prominence of the Hall term, j×B, in the generalizedOhm’s law, where j is the current density arising from unequalion and electron fluid velocities. To show this in the dynamicalevolution of the magnetic field, we will turn to the Hall-MHDmodel, as it is the simplest physical model of a plasma withunmagnetized ions (Shaikh & Shukla 2009). Although HallMHD might not be an accurate representation of the full kineticphysics, especially the linear kinetic physics in hot plasmas (Itoet al. 2004; Howes 2009), it serves as a simpler nonlinear modelto explain the rising magnetic compressibility and the isotropythat we observe here. Primarily, we will look at the Hall-MHDinduction equation (in SI units):

∂B∂t

= ∇ × (v × B) +1

μene

∇ × (B · ∇B) + η∇2B , (3)

where v is the bulk (ion) velocity, η is the magnetic diffusivity,μ is the permeability, e is the electron charge, and ne is theelectron number density. The first term on the right-hand side ofEquation (3) is the convective/dynamo term, the second term isthe Hall term, and the last term is the diffusive term. We haveneglected the effects of any electron pressure gradients here tosimplify the discussion, and as these are O(δne/n2

e) their effectsare considered small. We have kept the diffusive term—which isnormally negligible for these solar wind parameters—in orderto retain an energy sink in the equations, but also for thepossibility of a renormalized, or turbulent, magnetic diffusivity;however, we will not be making use of this term in the followingarguments. Although other terms, such as the electron inertiathat are very relevant to high-frequency modes, e.g., Whistlermodes, become relevant at scales kλe ∼ 1 (where λe is theelectron inertial length), we stick to the Hall-MHD model aboveand focus on the effects of the Hall term with respect to theconvective term. Importantly, in β ∼ O(1) plasmas (λi ∼ ρi asλi = ρi/

√β), such as the solar wind at 1AU, as one approaches

the ion-Larmor scale ρi , the Hall term becomes of the orderof the convective term (Goossens 2003). In the inertial range,at scales above the Larmor radius, it is the convective termwhich dominates and the magnetic field is frozen to the ion flow,such that any flow of the plasma perpendicular to the magneticfield affects the evolution of the magnetic field. Thus, if weassume that much of the power in inertial range magnetic fieldfluctuations is locally generated from the velocity fluctuations(via the nonlinear evolution of the ion momentum equation) inthe form of a dynamo effect (∇ × (v × B) term in Equation (3)),and is not being passively advected in the solar wind from theSun, then it is natural that most of the power in the inertial rangewill be in transverse fluctuations—also considered a signatureof Alfvenic fluctuations (Belcher & Davis 1971; Horbury et al.2005). Any residual parallel (compressible) fluctuations arisefrom fluctuations in the plasma density (which also contributeto the transverse fluctuations).

If we now approach scales close to ρi the Hall term starts toshow its effects. This manifests itself in changing the direction ofthe fluctuations such that if transverse fluctuations dominate, theHall term will generate parallel fluctuations and thus result in anincrease in the magnetic compressibility C‖. To see this we willfocus on the Hall term and assume an oversimplification that(1) it is entirely transverse fluctuations δB⊥ which dominatethe turbulent fluctuations coming from the inertial range and(2) that k⊥ fluctuations dominate (i.e., perpendicular gradients∇⊥)—both of which are supported by our observations andstudies of the full three-dimensional k-spectrum (Sahraoui et al.2010). This results in the Hall term contribution to the induction

equation becoming

∂B‖∂t

∣∣∣∣Hall

∼ ∇⊥1 × (B · ∇δB⊥2) + ∇⊥2 × (B · ∇δB⊥1) , (4)

where we have explicitly separated the two transverse compo-nents to make clear that the two directions are not parallel to eachother in the vector cross product. Equation (4) clearly shows theorigin of the enhancement in the parallel magnetic compress-ible fluctuations. In k-space the right-hand side of Equation (4)would be in the form of the following convolution:

∫d3j k ·B(k− j, t)(k⊥1 × δB⊥2(j, t) + k⊥2 × δB⊥1(j, t)), (5)

where the wavevector arguments are shown explicitly. Moreimportantly, Equations (4) and (5) also suggest the originof the isotropy that we observe among the different fluctua-tion components at scales close to ρe, where terms similar toEquations (4) and (5) are dominant and the convective termbecomes negligible. At such scales, the Hall term will domi-nate and will also convert any parallel fluctuations which aregenerated (or already present) into transverse ones. This willevolve until a steady state is reached between the various termscorresponding to the vector components of the magnetic fieldfluctuations, i.e., isotropy between all the vector components. In-terestingly, this simple observation of the role of the Hall term inthe enhancement of magnetic compressibility and the resultantisotropy, also suggests some information on the k-anisotropy,i.e., k‖/k⊥. If, say, Equation (5) was dominated by k‖ insteadof k⊥, then it is clear that an inertial range dominated by trans-verse fluctuations will transition into something which is alsodominated by transverse fluctuations, with little or no parallelfluctuations—this is not what we observe here. This indicatesthat, in our Hall-MHD formalism, power in k‖ fluctuations isnot dominant. Intriguingly, however, this type of argument alsodoes not rule out the possibility that both k‖ and k⊥ fluctuationsare significant at sub-Larmor scales (Gary et al. 2010); the onlyobservational study of the full three-dimensional P (k) spectraat such scales by Sahraoui et al. (2010) seems to indicate thatthis is not the case. This and more detailed calculations of theenergy transfer between wavevector triads of fluctuations willbe the subject of a future investigation, so we can determinea more precise nature of such steady states. Crucially, mea-surements of the magnetic compressibility, in the context of aHall-MHD model, could possibly constrain the measurementsof the k vector anisotropy.

3.1.3. The Transition Range

A transition range between the inertial and dissipation rangeswas suggested by Sahraoui et al. (2010), which was, in thecontext of dispersive waves, cited as a possible sign of wheremagnetic field energy is Landau damped into ion heating. Thistransition range, comprising just under a decade of scales aroundkρi ∼ 1 is also seen in the reduced spectra in Kiyani et al.(2009b) and Chen et al. (2010) and is distinctly different fromthe power law which follows at smaller scales down to kρe ∼ 1(see Figure 1 in Kiyani et al. 2009b and Figure 2 in Sahraouiet al. 2010). Our arguments above suggest that the competitionbetween the convective and Hall terms could also explain sucha transition range. In this case, and from looking at wherethe compressible fluctuations begin to rise in Figures 1 and 2(kρi ∼ 0.1), it seems that the transition range could actually

5


span a much larger range of scales. It is important to note herethat we do not know what the functional form of the transitionfrom the convective-dominated regime to the Hall-dominatedregime is. All we know is that in the inertial range, far fromthe spectral break, one can neglect the effects of the Hall term;well below the ion-Larmor radius and well past the spectralbreak we can neglect the convective term; and at kρi � 1 (forβi ∼ 1) these terms are of the same order as the other (Goossens2003). Using the above arguments, and the latter observationabout the similar strengths of the convective and Hall terms atkρi � 1, we can make the following observation: at kρi � 1 theconvective term will provide half of the fluctuation power (50%)which we assume, from the arguments and observations above,will be comprised of 10% fluctuations in the parallel direction;the Hall term will contribute the other half of the fluctuationpower (50%), and 33% of these fluctuations will be in theparallel direction. This means that the magnetic compressibilityat kρi � 1 will be

C‖ � 0.5 × 0.33 + 0.5 × 0.1 � 0.22. (6)

This value is in excellent agreement with the value of C‖ atkρi � 1 extrapolated from the local field curves in Figure 2.

3.2. Higher-order Statistics and Intermittency

We next calculate higher-order statistics given by the structurefunctions (absolute moments of the fluctuations; Kiyani et al.2009b) for the different components of the magnetic field fluc-tuations with respect to the local scale-dependent backgroundmagnetic field. The mth order wavelet structure function (Farge& Schneider 2006) is given by

Sm‖(⊥)(τ ) = 1

N

N∑j=1

∣∣∣∣δB‖(⊥)(tj , τ )√τ

∣∣∣∣m

, (7)

where, as detailed in the Appendix, τ = 2iΔ : i ={0, 1, 2, 3, . . .} is the dyadic timescale parameter related to thecentral frequency f used earlier, and Δ is the sampling period.Scale invariance is indicated by Sm

‖(⊥)(τ ) ∝ τ ζ (m), where ζ (m)are the scaling exponents. The structure functions and corre-sponding scaling exponents ζ (m) are shown in Figure 3 for boththe inertial and dissipation ranges using the ACE and Clusterintervals respectively.

Similar to the results by Kiyani et al. (2009b), the higher-order scaling in the inertial and dissipation ranges are distinct.The inertial range shows multiexponent scaling as evidencedby a nonlinear ζ (m) characteristic of solar wind turbulence atMHD scales (Tu & Marsch 1995). In contrast, the dissipationrange is monoscaling, i.e., characterized by a linear ζ (m) = Hmand a single exponent H. Notably, both parallel and transversefluctuations in the dissipation range show this different scalingbehavior in the inertial and dissipation ranges. However, thescaling behavior between the different components is alsodistinct, with a more pronounced difference shown in the inertialrange. The difference in the dissipation range exponents forparallel and transverse fluctuations is simply reflecting thedifferent spectral exponents seen earlier in Figure 1.

Before we discuss these higher-order scaling results, we com-plete the statistical results by finally looking scale by scale atthe individual PDFs for the transverse (e⊥1 direction) and par-allel fluctuations. It is necessary to pick one of the transversedirections as the combined (magnitude of) transverse fluctua-tions are positive-definite quantities, and thus do not illustrate

10

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ζ(m

)

parallelH=0.85transverseH=0.98

(a) (b)

(c) (d)

Figure 3. (a) Transverse and parallel wavelet structure functions of order 1–5(from the bottom) and (c) resultant scaling exponents for the inertial rangeusing the ACE data interval. Structure functions have been vertically shifted forclarity. (b) and (d) have descriptions similar to (a) and (c) but are from data inthe dissipation range using the Cluster data interval. These are the anisotropicgeneralizations of the differences in the scaling behavior between the inertialand dissipation ranges first shown by Kiyani et al. (2009b).


any symmetric/asymmetric character of the fluctuations. Thereis no a priori reason, from the symmetry of the physics, for onenot to expect the fluctuations to be axisymmetrically distributedin the plane perpendicular to the background magnetic field.However, if we look at the separate PSDs for the two trans-verse components of the magnetic field fluctuations (not shownhere) we will notice that this symmetry is broken and the powerin the two transverse components is distinctly different. Whenwe constructed our scale-dependent orthonormal bases (see theAppendix), it was natural to involve the background guide fieldas it is ubiquitously known to order the physics in magne-tized plasmas. The other two directions perpendicular to thisare relatively arbitrary, and we naturally chose the stable meanbulk velocity field direction, Vsw, to form these in the mannerof Belcher & Davis (1971): an orthonormal scale-dependent“field-velocity” coordinate system. However, the solar windbulk velocity field picks a preferred sampling direction (ink-space) resulting in the reduced spectrum mentioned earlierin the Introduction. In introducing the velocity field in such away, the reduced spectrum breaks the transverse axisymmetryof the magnetic field fluctuations and introduces a measure-ment bias (for further details, see Turner et al. 2011, who usecomparisons with MHD turbulence simulations, and a model oftransverse waves to show the importance of the spectral slope inthis broken symmetry). The magnitude of the transverse vectoris not affected by such a bias, so all our results above are ambiva-lent to this symmetry breaking. Although the PSD of the twotransverse components differs in this respect, we can confirmthat the standardized (rescaled) PDFs for both transverse com-ponents are nearly identical. Thus, the breaking of the transverseaxisymmetry does not affect the results presented in this paper.

The PDFs for δB‖ and δB⊥1 are shown in Figure 4,where we have used the self-affine scaling operationPs(δBiσ

−1) = σP (δBi, τ ) to rescale (standardize) the

6


−20 0 20−7

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−1 ] dissipation

rangeinertialrange

TransverseFluctuations

Figure 4. Rescaled (standardized) transverse and parallel PDFs of fluctuationsin the inertial and dissipation ranges (with Poisonnian error bars). Normalizedhistograms with 300 bins each were used to compute the PDFs. Four values ofτ were used in the inertial range and four different values of τ were used in thedissipation range: τ = {16, 32, 64, 128} s for the inertial range from ACE, andτ = {0.036, 0.071, 0.142, 0.284} s for the dissipation range from Cluster.


fluctuations by their standard deviation so as to offer a compari-son of the functional form of the PDFs for different τ . We showoverlaid rescaled PDFs corresponding to four values of τ in thedissipation range, τ = {0.036, 0.071, 0.142, 0.284} s (fromCluster), and in the inertial range, τ = {16, 32, 64, 128} s(from ACE). Figure 4 shows that in the dissipation range thesame PDF functional form is obtained over the range of τ forboth parallel and transverse fluctuations—suggesting that thedissipation range is in this sense “process isotropic” with a(speculatively) single physical process determining the dynam-ics. This supports our arguments and results above which showthe growing isotropy in the components of the magnetic fieldfluctuations. This is in contrast to the inertial range where we seethat the transverse and parallel PDFs are clearly different as wewould anticipate from their distinct ζ (m) shown in Figure 3(c).

In addition to the differences in the scaling behavior, both iner-tial and dissipation range PDFs are highly non-Gaussian indicat-ing high levels of intermittency. In contrast, the rescaled PDFsof the STAFF-SCM instrument noise shown in Figure 5 areclearly Gaussian, confirming that our results are robust to noisecontamination. Furthermore, we have analyzed other intervals,2007/20/01 12:00–13:15 UT and 2007/20/01 13:30–14:10 UT,and find broadly consistent results with those presented above(H‖ = 0.78, H⊥ = 0.95 and H‖ = 0.8, H⊥ = 0.94, respec-tively), but with lower signal-to-noise ratios. Please note thatwe consider “intermittency” in the most general sense to simplyreflect the presence of rare, or “bursty,” large amplitude fluctu-ations in our signals—something which is manifest in our non-Gaussian PDFs in Figure 4. This is in contrast to the populardefinition, advocated by Frisch (1995), which links intermit-tency to a dependency of the kurtosis κ(τ ) = S4(τ )/(S2(τ ))2 (oranother higher order moment ratio) with scale τ—a property ofmultifractal (multiexponent) scaling, i.e., as seen for the inertialrange in our results. In this latter definition, the constancy ofthe kurtosis with scale indicates “self-similar” signals such as

Figure 5. Rescaled PDF of Bz magnetic fluctuations from the instrument noiseproxy of Figure 1, at the same scales τ as in Figure 4 for the dissipation range.A maximum likelihood estimate clearly shows the Gaussian nature of this noiseproxy—very different from the highly non-Gaussian statistics shown in Figure 4.We can also confirm that the smallest scales in our data which coincide withthe noise floor, in terms of power and Gaussian statistics, are excluded from thestudy presented in this article.


our monoscaling signature seen in the dissipation range. Thereis no need to plot the kurtosis for our data as this can be veryeasily seen if we use the scaling relationship of the mth-orderstructure function Sm(τ ) = τ ζ (m)Sm(1) (see Kiyani et al. 2006for notation) in the definition for the kurtosis. This will resultin κ(τ ) = τ ζ (4)−2ζ (2)κ(1), which, if ζ (m) is a linear functionof m (ζ (m) = Hm), is independent of τ , as is the case forour results in the dissipation range. However, even though ourresults show a monoscaling signature in the dissipation range,there is still a small nominal dependence on the scale τ for κ(τ ).This could be simply due to a statistical artifact associated withfinite sample size and the fact that the kurtosis is sensitive tovery large fluctuations. Our very largest events at the tails of ourdistribution are not statistically well-sampled—an unavoidablepitfall of heavy-tailed distributions, as seen by the large errorsat the tails of our PDFs in Figure 4.

3.2.1. Comparison with Other Works on Scaling at kρi > 1

We now compare our results for higher-order statistics in thedissipation range to other studies from both observations andsimulations. In the case of the dissipation range, there are onlyreally two of these: the study of Alexandrova et al. (2008) whoused normal-mode (25 Hz) Cluster observations and Cho &Lazarian (2009) who conducted 5123 electron MHD (EMHD)simulations. Although EMHD is a high-frequency model formagnetic field dynamics with inertialess ions (the oppositeof Hall MHD), within the scales that we are analyzing here,ρ−1

i � k � ρ−1e , the induction equation for both Hall MHD and

EMHD are nearly identical for an order unity βi plasma. Thus,the authors (Cho & Lazarian 2009) also use EMHD to study thedynamics and statistics of magnetic field fluctuations in solarwind turbulence.

Alexandrova et al. (2008) used a continuous wavelet trans-form using the Morlet wavelet to calculate both PDFs and flat-ness (kurtosis) functions. They too found non-Gaussian PDFswithin the dissipation range but, in contrast to our monoscaling

7


results, showed that the dissipation range has a kurtosis whichincreases rapidly with scale τ , i.e., suggestive of multifractalscaling. They also showed a dependence of this flatness func-tion with βi . The increase of compressible fluctuations coincidedwith a steepening of the spectral slope and the increase of in-termittency, as reflected in the flatness function. This promptedthe authors to suggest that another nonlinear energy cascade,akin to the inertial range, takes place in the dissipation range.However, due to the increased role of compressible fluctuationsat these scales, driven by plasma density fluctuations, this is asmall-scale compressible cascade. Due to the dominance of theHall term at these scales, Alexandrova et al. (2008) then sug-gest a simple model based on compressible Hall MHD, whichtakes into account density fluctuations to describe the deviationof the energy spectrum slope from the −7/3 predicted purelyby the induction equation for incompressible EMHD (Biskampet al. 1996). Without a further study of more data intervals, itis difficult to say why our interval shows quite a different scal-ing behavior to the interval investigated by Alexandrova et al.(2008). However, considering our results and our earlier argu-ments regarding the magnetic compressibility and the role of theHall term, the notion of a compressible cascade is an appealingone. Although specific results were not explicitly shown, weshould also mention the work of Sahraoui & Goldstein (2010),where the monoscaling result presented in this article and inKiyani et al. (2009b) was also independently confirmed withCluster magnetic field data.

The very informative article by Cho & Lazarian (2009)describes the results of their EMHD turbulence simulations,as well as ERMHD simulations. The UDWT digital filters usedin our study are similar (in spirit) to the multipoint structurefunctions used in the analysis of Cho & Lazarian (2009). Themain feature of these simulations which is relevant to ourresults is that, for the EMHD simulations, the monoscalingbehavior found in our results is also seen in the higher-orderstatistics computed from multipoint structure functions formoment orders up to m = 5. Computing higher order momentsgreater than 5 becomes very difficult in non-Gaussian heavy-tailed statistics due to the poor sampling in the tails (DudokDe Wit 2004). The scaling exponent H for their ζ (m) = Hmscaling (as they used ζ (p)/ζ (3)) is not explicitly stated intheir paper. However, this is easily obtained from the −7/3spectral slope that they obtain in their energy spectrum (forboth EMHD and ERMHD), using the well-known relationshipβ = −(2H + 1) (Kiyani et al. 2009b), where β is the spectralslope. We can then infer that H = 2/3 � 0.66. This isthe scaling for the r⊥ (k⊥) variations. The results of Cho &Lazarian (2009) also show moderately non-Gaussian PDFs.The authors then compute two-point flatness functions, as wellas wavelet flatness functions using the Morlet wavelet similarto Alexandrova et al. (2008). Interestingly, for r⊥ they finda nominal dependence of the flatness with separation scaler (equivalent to τ ) for the two-point flatness function, but alarger increase with r for the wavelet flatness function. Thislast part offers slightly conflicting results when consideringtheir multipoint structure function monoscaling for momentsof order less than 5. Nevertheless, the results of Cho &Lazarian (2009) offer the closest agreement to our scaling resultspresented in this article and serve as a good comparison to anappropriate model for dissipation range fluctuations (EMHDinduction equation as opposed to ideal MHD). Similarly toAlexandrova et al. (2008), we could speculate that the differencein our exponents, H⊥ = 0.98 and H‖ = 0.85 (and also the

respective spectral exponents), and the −7/3 from EMHD couldbe due to the differing role of density fluctuations. However, inrecent gyrokinetic simulations of anisotropic plasma turbulence,Howes et al. (2011b) show that spectra steeper than −7/3 areonly obtained if the full kinetic physics is considered in termsof retaining a physical damping mechanism, e.g., collisionlessLandau damping, transit time damping etc. This is absent in theEMHD, ERMHD, and Hall-MHD descriptions (see also Howes2009). The gyrokinetic description retains this missing physicsand thus Howes et al. (2011b) claim that their simulationssuccessfully recover the steeper scaling seen here and in Kiyaniet al. (2009b), Alexandrova et al. (2009), Sahraoui et al. (2010),and Chen et al. (2010) for similar plasma parameters.

3.2.2. Data Uncertainties and Errors

There is a small but important data caveat that one shouldmention with regards to these results. This concerns the noisefloor of the STAFF search-coil magnetometer and the signal-to-noise ratio at small scales near the electron gyroradius.Due to the dyadic frequency spacing of the UDWT that weuse, frequency resolution is poor. We pay this expense forthe faster algorithms and the smoother spectra that we gain,the latter being preferred for broadband spectra where robustestimates of the spectral slope are needed. This gives theimpression that the signal-to-noise ratio at the smaller scalesis higher than it might actually be. In reality there exist somehigh power spikes in the noise-floor signal >60 Hz whichcould contaminate the observations at the electron gyro-scale(see Figure 1 in Kiyani et al. 2009b). These spikes are dueto interference from electrical signals of the Digital WaveProcessor (DWP) instrument on board Cluster (N. Cornilleau-Wehrlin 2012, private communication). The DWP signals areused to synchronize instrument sampling within the experimentsof the Cluster Wave Experiment Consortium (WEC), whichSTAFF belongs to. These spikes only affect the last data pointin Figures 1 and 2 and thus should not change our results andconclusions significantly. Intervals with higher power such asthose in the slow solar wind stream would be needed for betterestimates at these higher frequencies. In addition, the noise-floor that we use is not the actual noise-floor but a proxy forit, obtained from a very quiet period in the magnetotail lobeswhere the magnetic field power is very low. The actual noise-floor could be smaller. A discussion on the noise-response ofthe STAFF search-coil instrument and its sensitivity in differentplasma conditions can be found in Sahraoui et al. (2011).

We would also like to note that the PDF for the parallelfluctuations seen in Figure 4 for the inertial range from ACE,show an unusual skew toward positive fluctuations. This isan artifact for this particular interval of ACE magnetic fielddata due to large spike-drops (discontinuities) in the magneticfield magnitude |B|. As these spikes arise in |B| and not in thecomponents, this reflects itself more in the parallel componentof the fluctuations rather than the transverse components as, inmany cases, one can use |B| as a proxy for parallel fluctuations,e.g., in Alexandrova et al. (2008). This is also reflected in thelarge errors for the parallel scaling exponents in the inertial rangeshown in Figure 3(c). However, as these spikes are primarilyin the large rare events in the positive tail of the PDF, webelieve that the differences in the PDFs of the transverse andparallel fluctuations within the inertial range still occur and ourdiscussion above remains valid.

8


3.2.3. Discussion

The isotropy that we observe between the power in thedifferent fluctuation components at kρe ∼ 1 is by no meansa result which would apply to all plasma environments. Indeed,there is some strong observational evidence that the ion/protonplasma βi also has a strong role to play here. This was shownby Hamilton et al. (2008), who repeated the inertial rangeanalysis of Smith et al. (2006b) within the dissipation rangeand showed that the variance anisotropy followed the empiricalpower-law relationship δB2

⊥/δB2‖ ≡ PSD⊥/PSD‖ ∼ β−0.56

p

in open field line intervals of the solar wind from the ACEspacecraft. However, these results were limited to 3Hz cadenceand thus could not sample as much of the dissipation rangeas in this article. Also Hamilton et al. (2008) used a globalfield (mean field over the entire interval) rather than thelocal scale-dependent field used in this article. This samedependence of the magnetic compressibility on βi was alsoindicated in higher cadence (25 Hz) Cluster data in the workby Alexandrova et al. (2008), again suggesting that higher βi

implies higher levels of C‖ (these authors used δ|B| as a proxyfor δB‖). The effect of varying βi is not mentioned explicitlyin our arguments using the Hall term explained above, as wehave neglected density fluctuations to simplify the discussion.However, purely from the induction equation point of view, βi

is taken into account in dissipation range scales if we retainthe effects arising from density fluctuations at scales kρi > 1as in the formulation of electron-reduced MHD (ERMHD;Schekochihin et al. 2009; Cho & Lazarian 2009). In ERMHD,the explicit βi dependence is shown, assuming pressure balance,in the pre-factor of the ∇⊥ × (B · ∇B⊥) term in the inductionequation for B‖. Thus, this addition of density fluctuationscan generalize our arguments above which were essentiallybased on the Hall term alone. At higher or lower than unityβi , this will break the isotropy of the fluctuation componentsseen in this article at kρe ∼ 1, as an additional anisotropicsource of magnetic compressibility will be introduced via thedensity fluctuations (see also Malaspina et al. 2010 for high-cadence measurements of density fluctuations in the solar windbetween ρ−1

i < k < ρ−1e ). The explicit βi dependence for

the rising compressibility is also seen in the fluid version ofthe kinetic Alfven wave solution of compressible Hall MHDwhere the compressible component to the linear solution goes asδB‖ ∼ kλi(βi + 1 + sin2 θkB)/ sin θkB (Sahraoui 2003), whereλi is the ion inertial length and θkB is the angle between thewavevector k and the magnetic field vector B. This solutionassumes that sin θkB �= 0, thus it is not valid for purely parallelpropagating Alfven waves. However, the rising compressibilityis clearly seen here as k → λi .

Note that in most of the previous arguments we have beenslightly cavalier in our discussion of field components. Ourintroduction of the local background magnetic field should notbe looked upon lightly. It is clear that in cases where δB � B0,where B0 is the global background magnetic field, both globaland local field descriptions should have negligible differences.However, as we observed in the section describing higher-orderstatistics, it is not obvious that this is the case here, as thedistribution of δB is highly non-Gaussian with very heavy tails.In our explanation of the rising compressibility in sub-Larmorscales using the Hall-term and also the lower levels of thecompressibility in the inertial range using the convective termin the induction equation, we did not mention the spatial (andpossibly temporal) gradients which arise due to the now locally

varying background magnetic field. In a global DC backgroundfield, or very slowly evolving field, these gradients are null ornegligible. If there is a large spectral separation between thescales of the background magnetic field and the fluctuationsthat we are projecting on it, then this conclusion could still bequite valid. However, as our UDWT method ensures that nosuch large gap exists, the statistics we calculate will include theeffects of such gradients. Further investigation and mathematicalgrounding of a local-field is of crucial importance, as the notionof using a local versus a global background magnetic fieldis intimately tied with the issue of whether we have a strongbackground guide field or a weaker more stochastic one (see thediscussion by Schekochihin et al. 2009 and references therein forfurther details and exploration of this topic; and also the recentdiscussion by Matthaeus et al. 2012 on the stochastic natureof the local background field and its relationship with globalbackground field statistics). This also brings into question theuse of linear wave modes, such as the kinetic Alfven wave,which are normally described as small amplitude perturbationson a static or slowly evolving equilibrium field. Although finiteamplitude propagating nonlinear Alfven waves are an exact(Elsasser) solution to the ideal MHD equations and similararguments in the context of the ERMHD equations are made bySchekochihin et al. (2009) for finite amplitude kinetic Alfvenwaves, this topic also requires further exploration (see also theexcellent discussion of this by Dmitruk & Matthaeus 2009, whoexplore the importance of the strength of the mean backgroundfield using direct numerical simulations of the incompressibleMHD equations).

One might question a conclusion of a single physical processor type of mediating dynamics (in the dissipation range) by theobservation that both transverse and parallel components haveslightly different scaling exponents and spectral indices. Thiscan be reconciled by realizing that there is no contradiction in thefact that the two components can have slightly different scalingexponents but have identical PDF functional forms. Take forexample the case of fractional Brownian motion (fBm), which isa non-Markovian generalization of Brownian motion. Two fBmstochastic processes with different scaling exponents will stillhave the same Gaussian functional form for their fluctuations. InfBm the scaling exponents are simply an additional parameterrepresenting the degree of statistical dependence between thefluctuations (Mandelbrot 1983; Samorodnitsky & Taqqu 1994).

If we look from the point of view of linear wave modecharacteristics, the distinction between inertial and dissipationrange scalings could also be seen to reflect the decoupledand coupled nature of the wave modes in the inertial rangeand dissipation ranges respectively (Leamon et al. 1999).The decoupling of these modes in the inertial range alsoresults in the decoupling between transverse and compressiblemagnetic field fluctuations (Schekochihin et al. 2009). In thispicture, the scaling in the dissipation range could then bedescribed by nonlinearly interacting kinetic Alfven waves. Inall these cases, as they involve linear wave modes, the nonlinearcascade would then take the form of “critical-balance” typecascades as in the works of Schekochihin et al. (2009) andHowes et al. (2008, 2011b), or if one can show that thenonlinearity is small, a “weak-turbulence” type description(Galtier et al. 2002). Although there is no justification forthe latter in the inertial range, there might be a case for it inthe dissipation range if the coupling between modes becomesweak and results in small amplitude fluctuations (Rudakovet al. 2011). In hydrodynamics described by the Navier–Stokes

9


equations, this latter assumption can be valid if one uses a localReynolds number (see Batchelor 1953), which is small in thehigh-k regime (when the spectral amplitudes become small)well into the hydrodynamic dissipation range. This can thenprovide a suitably small parameter to base a convergent (orasymptotic) perturbation expansion. Of course, without concretemeasurements of nonlinearity, this is all speculation. However,it could possibly be an avenue for a future (maybe fruitful)investigation.

An open question is why such a transition from multifractalscaling to self-similar monoscaling occurs between the inertialand dissipation ranges. Kiyani et al. (2010) briefly discussthis using hydrodynamic analogies (Frisch & Vergassola 1991;Chevillard et al. 2005) of a near-dissipation range where theincreased effect of viscosity successively switches off theavailable scaling exponents of the multifractal field—equivalentto narrowing the “f (α)” spectrum in multifractal models.However, no such apparent analogies with a viscous “fractal-dampening” can happen in a near-collisionless solar wind. Also,unlike the hydrodynamic analogy this transition happens veryfast as we cross kρi ∼ 1, although a scaling analysis withinthe “transition range” might yet reveal an intermediate scalingbehavior which shows this successive “switching-off” of scalingexponents. Indeed, the steepening of the spectral slope in thetransition region is strongly suggestive that such an intermediatescaling range could possibly exist. This is an exciting prospectand will be the subject of a future investigation. This latterinvestigation is not feasible with our UDWT technique as dueto the dyadic spacing, it has poor frequency resolution—amore detailed “fine-toothed comb” technique such as the useof wavelet packets (Walden & Cristan 1998) or the continuouswavelet transform (Horbury et al. 2008; Podesta. 2009) wouldbe needed here.

4. CONCLUSIONS

In this article, we have performed a systematic scale-by-scale anisotropic decomposition of magnetic field fluctuationsin the solar wind across both the inertial and dissipation ranges,from MHD to electron scales. We use a novel scheme based ondiscrete wavelet filters which self-consistently decomposes themagnetic fields into scale-dependent background and fluctuatingfields. From the background fields we then construct a localorthonormal scale-dependent field-velocity coordinate system,on which we then project the fluctuations and calculate ourstatistical quantities. Our main findings, based on the one hourhigh-cadence burst-mode data from Cluster, are as follows.

1. There exists a successive scale-invariant reduction in thepower ratio between parallel and transverse components aswe move to smaller scales below the ion-Larmor radius ρi .Importantly, at kρe � 1 with k components highly oblique/perpendicular to the background magnetic field, the fluctua-tions become isotropically distributed between the power inall three components (two transverse and one parallel), i.e.,equipartition of magnetic energy between all components.

2. By comparing with linear solutions of the Vlasov equationand computing the magnetic compressibility we showthat the scale-invariant reduction in the power ratio isalso qualitatively consistent with the transition to kineticAlfven wave solutions which rapidly develop magneticcompressible components at scales around ρi .

3. Using a Hall-MHD model we provide a simple (nonlinear)description of this new result in terms of how the Hall

term is responsible for the rise of parallel (compressible)magnetic fluctuations and the resultant isotropy at scalesaround ρe.

4. Higher order statistics and PDFs for parallel/transversefluctuations reveal that the dissipation range is charac-terized by a single monoscaling statistical signature withan isotropic distribution of fluctuations, thus supportingthe above observations of isotropy. This is in contrast tothe inertial range where the corresponding signature isanisotropic and multiscaling (multifractal).

In the compressible Hall-MHD framework, at such scalesthe velocity field is expected to decouple from the evolutionof the magnetic field and some researchers (Servidio et al.2007; Alexandrova et al. 2008) have suggested that a newnonlinear compressible cascade is responsible for the newpower-law regime in the PSD. In this context, we should alsomention that all of the arguments presented in this paper donot include any discussion of the possibility of phenomenaarising from the effects of (turbulent) reconnection (Lazarianet al. 2012) or of the possible role of a turbulent or renormalizedmagnetic diffusivity—of which either, or both, are purportedto be important at such scales. Neither have we discussed therole of density fluctuations, which have been neglected in ourarguments, but are shown to play a possibly significant rolethrough the electron compressibility (Gary & Smith 2009).Future work would involve looking at the possible importanceof such fluctuations as well as investigating the energy transferbetween modes in Hall-MHD/Electron MHD theories.

In the context of theories of turbulence which invoke plasmawave modes as the mediators of the energy cascade, two candi-date wave modes are classically suggested in the dissipationrange: quasi-perpendicular kinetic Alfven waves and quasi-parallel Whistler waves. As first suggested by Gary & Smith(2009) and shown in Salem et al. (2012), the magnetic com-pressibility, C‖, might be a more robust measurement to dis-tinguish between the different wave modes possibly mediatingthe energy cascade, than the phase speed plots calculated byE/B ratios (Bale et al. 2005; Sahraoui et al. 2009). In hot plas-mas (βi � 2) and at highly oblique angles to the backgroundmagnetic field, classical Whistlers arising as an extension ofthe fast magnetosonic modes at high frequencies �ωci (whereωci is the ion cyclotron frequency) are strongly damped andsplit up into different ion Bernstein modes. Recently, however,Sahraoui et al. (2012) have shown that, under the same condition,a Whistler-like branch having similar properties (e.g., dispersionand polarization) as the classical Whistler mode rises from thekinetic Alfven wave mode and extends it to very high frequen-cies ω � ωci . Also, if there is a significant amount of power inshallow angles, e.g., 30◦, between k-vectors and the backgroundmagnetic field, then the classical quasi-parallel Whistlers couldalso show the same amount of magnetic compressibility that weobserve in this paper (Gary & Smith 2009; Salem et al. 2012).

Our study of higher order statistics show a new and significantresult of the change of anisotropy at ρi . As in the behavior ofthe inertial range, these scalings and statistics are a distinctiveand important characteristic of the turbulence at sub-ion Larmorscales and any cascades which take place there. Therefore, it isessential that any comprehensive statistical theory of plasmaturbulence for such scales should be able to describe suchstatistics.

An open question is whether these results depend upon bulkplasma parameters such as βi and in particular on the angle ofk to the background magnetic field. A limitation of the current

10


work is that it is based on a handful of data sets with verysimilar plasma parameters. A significant and maybe conclusiveadvance would be made if a more comprehensive survey wasconducted. This would undoubtedly be a fruitful avenue forfuture investigations.

The authors acknowledge the ACE Science Centre; Wind 3DPand MFI teams; Cluster instrument teams for FGM, STAFF-SC, CIS, WHISPER, and PEACE; and the DSP group at RiceUniversity for use of the Rice Wavelet MATLAB Toolbox.K.K. acknowledges C. Chen, K. Osman, T. Dudok De Witt,P. Gary, and G. Howes for useful discussions and advice on thetechniques and physics in this article. O.F. thanks the Comms.& Signal Processing group in the Department of Electrical andElectronic Engineering, Imperial College London, for hostinghim as a visiting student. K.K., S.C., and F.S. are members ofISSI team 185 “Dissipation/Dispersion in Plasma Turbulence,”in the context of which some of the work here was conducted.This work was supported by the UK STFC and the EPSRC.

APPENDIX

ORTHONORMAL SCALE-DEPENDENT BASIS AND THEUNDECIMATED DISCRETE WAVELET TRANSFORM

We use the UDWT method (also known as the stationary,translation-invariant, redundant, or a’trous wavelet transform(Nason & Silverman 1995; Ogden 1997; Mallat 2009)) todecompose the vector magnetic field observations into scale-dependent background and fluctuating components. Unlike thestandard discrete wavelet transform (DWT), the UDWT doesnot downsample the data at each stage of the transform. Thus,the UDWT provides information about the signal at eachobservation time and so retains event information (Walden &Cristan 1998; Percival & Walden 2000). This timing informationis necessary to project the fluctuations onto a mean backgroundfield at each observation time. However, the extra redundancydue to no downsampling means that the orthogonality propertiesof the DWT are no longer present in the UDWT.

In this article, we use the Coiflet 2 wavelet (coif2), whichprovides a 12-tap high/low pass filter pair, chosen due to thecompromise between time-frequency compactness, smootherspectral index estimate, and as its first four moments are zeroit captures spectral indices as steep as −9 (Farge & Schneider2006). This latter property is very important as the standardmethod of using increments (equivalent to using a Daubechiesorder 1 (db1), or Harr, wavelet with only one zero moment) tocalculate two-point structure functions in turbulence is limitedto spectral slopes shallower than −3 (Farge & Schneider 2006;Cho & Lazarian 2009). This is adequate for studying the inertialrange of turbulence where the spectral slope is ubiquitouslygreater than −2 (Salem et al. 2009), but is inadequate in thedissipation range where spectral slopes have been observed(Leamon et al. 1998a; Smith et al. 2006a) and theoreticallypredicted (Schekochihin et al. 2009) to have much steeper slopesbetween −3 and −5. Crucially, for data analysis purposes, thefilters corresponding to the coif2 wavelet are virtually symmetricand thus possess a virtually linear phase response (Daubechies1992). The coif2 wavelet (and corresponding scaling function)filters used here are identical to the ones published in Daubechies(1992).

The UDWT scheme presented below is similar to the wavelettechnique pioneered by Horbury et al. (2008) who used acontinuous wavelet transform (CWT), where the background

a1 d1

frequency

pow

er

a2 d1

frequency

pow

er

d2

details(small scales)

approximations(large scales)

Figure 6. Upper panel: a schematic, in terms of spectral frames, showing thefirst stage of a UDWT decomposition in frequency space: the raw signal isdecomposed into a low-pass (approximation), “a1,” and a high-pass (detail)component, “d1,” using the coif2 wavelet filters (low–high pass quadraturemirror filter pair). Each of these components has an equal (half) share of thefrequency space. Lower panel: at the next stage, the “a1” low-pass filtered signalis then further decomposed into a low-pass signal, “a2,” and a high-pass signal,“d2” (again equal share of frequency space). In this way the signal is decomposedinto dyadic frequency bands—equivalent to the application of a dyadic filterbank of bandpass filters to the signal. The resultant set of “details” {d1,d2,d3...}provides the fluctuations, and the set of “approximations” {a1,a2,a3...} (onceinverse transformed) provides the respective background fields to be projectedupon. The stars represent the central frequency of the resultant bandpass filteredsignals used for the fluctuations.


magnetic field was defined as convolutions with the scale-dependent envelope of the Morlet wavelet used. However, themethod described here differs from the CWT scheme used byHorbury et al. (2008), as the DWT and UDWT filters used areexplicitly designed self-consistently to provide both fluctuationsand background fields such that no information is lost and thesignal can be reconstructed exactly—the so-called quadraturemirror filter (Daubechies 1992). Thus, these filters also ensurethat there is no spectral gap between the background field andfluctuations. Details of the actual fast pyramidal algorithms usedto implement the UDWT can be found in Percival & Walden(2000) and Mallat (2009). For this article, we simply attempt tosummarize what are the salient outcomes of the technique. Thus,the description will be necessarily brief with limited details.A detailed description of the technique as applied to plasmaturbulence anisotropy, as well as an extension of this to non-dyadic wavelet packets, will be described in a longer article.

The UDWT is equivalent to the following operation. Ateach temporal scale τ (corresponding to a frequency) wesuccessively apply a pair of high- and low-pass wavelet filterson the data: these provide the “details” (fluctuations) and the“approximations” (precursor to background field). Since theseare dyadic filters, they divide the available frequency space inhalf at each stage. These filters are then upscaled by two and theprocess is then repeated on the previous approximation signal.The results of the first two stages of this process are shownin Figure 6. Importantly, at each stage of the UDWT, it is theapproximations that are being decomposed into a further setof approximations and details. The effective end result of thissuccessive filtering is a series of dyadic bandpass filtered signals

11


of the magnetic field B(tj ) (sampled at discrete times tj) to give afluctuation in terms of the wavelet coefficient δB(tj , τ ) (at timetj based on a temporal scale τ ):

δB(tj , τ ) =N∑

k=1

B(tk)[√

τψ(τ tk − tj )]

, (A1)

where τ = 2iΔ : i = {0, 1, 2, 3, . . .} is the dyadic scaleparameter, Δ is the sampling period of the observations, andψ is the effective bandpass filter comprised of a succession ofthe low-pass and high-pass filters. Note that without the factor√

τ and choosing the db1 (Harr) wavelet instead of the coif2, thedefinition in Equation (A1) is identical to calculating incrementswhich are used in structure function analysis of intermittencystudies in turbulence (Katul et al. 2001; Farge & Schneider2006; Salem et al. 2009). Note also that the resultant waveletcoefficients δB(tj , τ ) have to be phase-corrected due to thelinear phase shift introduced by the filtering (Walden & Cristan1998). Also, edge artifacts due to incomplete convolutions ofthe signal with the filters are discarded before the constructionof the “approximation” and “detail” signals.

The scale τ can be related to a central frequency f in Hz(Abry 1997) of a dyadic frequency band and so we can writeδB(tj , f ). The result of the low-pass filter at each stage cansimilarly be written as B(tj , f ) using the wavelet conjugatescaling function (Ogden 1997). It is important to note thatboth δB(tj , f ) and B(tj , f ) live in a function space spannedby the bases constructed from shifts and dilations of the coif2wavelet and scaling functions (i.e., shift tj and dilation τ as inEquation (A1)). We use the inverse UDWT operation on B(tj , f )in order to obtain the mean background field B(tj , f ), in thedirection of which we then project the fluctuations δB(tj , f ).

The mean field direction unit vector is then defined ase‖(tj , f ) = B(tj , f )/|B(tj , f )|. This is distinct from themean field coordinate presented in Belcher & Davis (1971)and Leamon et al. (1998b) as the mean field here is a lo-cally varying scale-dependent field consistent with the scale-dependent fluctuations. The other two perpendicular axesare e⊥1(tj , f ) = (e‖(tj , f ) × 〈Vsw〉)/|e‖(tj , f ) × 〈Vsw〉| ande⊥2(tj , f ) = e‖(tj , f ) × e⊥1(tj , f ), where 〈Vsw〉 is the solarwind velocity direction unit vector time-averaged over the en-tire interval. Since the solar wind velocity is in a very fastand steady stream, and is within ∼3◦ of the GSE x direc-tion in both the ACE and Cluster intervals, it is reasonableto take a time-averaged global, rather than a local, velocityfield. Together [e⊥1(tj , f ), e⊥2(tj , f ), e‖(tj , f )] form a time-and scale-dependent orthonormal basis. The fluctuations paral-lel δB‖(tj , f ) and transverse [δB⊥1(tj , f ) , δB⊥2(tj , f )] to themean field are then given by projections onto this new basis.

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enhanced magnetic compressibility and isotropic scale ... · tuations in this sub-ion larmor scale...

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